Massachusetts
Institute
of
Technology
Department
of
Electrical
Engineering
and
Computer
Science
6.243j
(Fall
2003):
DYNAMICS
OF
NONLINEAR
SYSTEMS
by
A.
Megretski
Problem
Set
3
Solutions
1
Problem
3.1
Find
out
which
of
the
functions
V
:
R
2
�
R
,
2
(a)
V
(
x
1
, x
2
)
=
x
2
+
x
2
;
1
(b)
V
(
x
1
, x
2
)
=

x
1

+

x
2

;
(c)
V
(
x
1
, x
2
)
=
max

x
1

,

x
2

;
are
valid
Lyapunov
functions
for
the
systems
(1)
x
˙
1
=
−
x
1
+
(
x
1
+
x
2
)
3
,
x
˙
2
=
−
x
2
−
(
x
1
+
x
2
)
3
;
2
2
2
2
(2)
x
˙
1
=
−
x
2
−
x
1
(
x
1
+
x
2
),
˙
x
2
=
−
x
1
−
x
2
(
x
1
+
x
2
);
(3)
x
˙
1
=
x
2

x
1

,
x
˙
2
=
−
x
1

x
2

.
The
answer
is:
(b)
is
a
Lyapunov
function
for
system
(3)
 and
no
other
valid
pairs
System/Lyapunov
function
in
the
list.
Please
note
that,
when
we
say
that
a
Lyapunov
function
V
is
defined
on
a
set
U
,
then
we
expect
that
V
(
x
(
t
))
should
nonincrease
along
all
system
trajectories
in
U
.
In
the
formulation
of
Problem
3.1,
V
is
said
to
be
defined
on
the
whole
phase
space
R
2
.
Therefore,
V
(
x
(
t
))
must
be
nonincreasing
along
all
system
trajectories,
in
order
for
V
to
be
a
valid
Lyapunov
function.
To
show
that
(b)
is
a
valid
Lyapunov
function
for
(3),
note
first
that
system
(3)
is
defined
by
an
ODE
with
a
Lipschitz
right
side,
and
hence
has
the
uniqueness
of
solutions
property.
Now,
every
point
(
x
1
, x
2
)
≤
R
2
with
x
1
=
0
or
x
2
=
0
is
an
equilibrium
of
(3).
Hence
V
is
automatically
valid
at
those
points.
At
every
other
point
in
R
2
,
V
is
1
Version
of
October
10,
2003
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2
differentiable,
with
dV/dx
=
[sgn(
x
1
);
sgn(
x
2
)]
being
the
derivative.
Hence
≡
V
(
x
)
f
(
x
)
=
x
1
x
2
−
x
1
x
2
=
0
at
every
such
point,
which
proves
that
V
(
x
(
t
))
is
nonincreasing
(and
nondecreasing
either)
along
all
nonequilibrium
trajectories.
Below
we
list
the
“reasons”
why
no
other
pair
yields
a
valid
Lyapunov
function.
Of
course,
there
are
many
other
ways
to
show
that.
For
system
(1)
at
x
=
(2
,
0),
we
have
x
˙
1
>
0,
x
˙
2
<
0,
hence
both

x
1

and

x
2

are
increasing
along
system
trajectories
in
a
neigborhood
of
x
=
(2
,
0).
Since
all
Lyapunov
function
candidates
(a)(c)
increase
when
both

x
1

and

x
2

increase,
(a)(c)
are
not
valid
Lyapunov
functions
for
system
(1).
For
system
(2)
at
x
=
(0
.
5
,
−
0
.
5),
we
have
x
˙
1
>
0,
x
˙
2
<
0,
hence
both

x
1

and

x
2

increase
along
system
trajectories
in
a
neigborhood
of
x
=
(0
.
5
,
−
0
.
5).
2
2
For
system
(3)
at
x
=
(2
,
1),
we
have
x
˙ =
(2
,
−
2),
hence
both
x
1
+
x
2
and
max(
x
1
, x
2
)
are
increasing
along
system
trajectories
in
a
neigborhood
of
x
=
(2
,
1).
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 Spring '09
 AlexandreMegretski
 Dynamics, X1, Stability theory, Lyapunov, Lyapunov function, x2 /2

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